A New Kind of Science: The NKS Forum

All Liar, No Paradox

According to my understanding of it,
the so-called Liar Paradox is just the
most simple-minded of fallacies, involving
nothing more mysterious than the acceptance
of a false assumption, from which anybody can
prove anything at all.

Let us contemplate one of the shapes in which
the re*putative Liar Paradox is commonly cast:

Somebody writes down:

1. Statement 1 is false.

Then you are led to reason:
If Statement 1 is false then
by the principle that permits
the substitution of equals in
a true statement to obtain
yet another true statement,
you can derive the result:

"Statement 1 is false" is false.
Ergo, Statement 1 is true,
and so on, and so on,
ad nauseum infinitum.

Where did you go wrong?
Where were you misled?

As it happens, graphical reasoning does help
to clear this up -- at least, it did for me --
if only because the process of translating
the purported reasoning into another form
of representation gave me a crucial clue
as to where the wool was being pulled.

Just here, to wit, where it is writ:

1. Statement 1 is false.

What is this really saying?
Well, it's the same as writing:

Statement 1. Statement 1 is false.

And what the heck does this dot.comment say?
It is inducing you to accept this identity:

"Statement 1" = "Statement 1 is false".

That appears to be a purely syntactic indexing,
the sort of thing you are led to believe that
you can do arbitrarily, with logical impunity.
But you cannot, for syntactic identity implies
logical equivalence, and that is liable to find
itself constrained by iron bands of logical law.

And you cannot, not with logical impunity, assume the result
of this transmutation, which would be as much as to say this:

"Statement 1" = "Negation of Statement 1"

To write down the last step in the form that I like:

(( Statement_1 , ( Statement_1 ) ))

And this my friends, call it "Statement 0",
is purely and simply a false statement,
with no hint of paradox about it.

Statement 0 was slipped into your drink
before you were even starting to think.
A bit before you were led to substitute
you should have examined more carefully
the site proposed for the substitution!

For the principle that you rushed to use
does not permit you to substitute unequals
into a statement that is false to begin with,
not just in the first place, but even before,
in the zeroth place of argument, as it were,
and still expect to come up with a truth.

Now let that be the end of that.

Jon Awbrey

Posted by: Philip Ronald Dutton

According to my understanding of it,
the so-called Liar Paradox is just the
most simple-minded of fallacies, involving
nothing more mysterious than the acceptance
of a false assumption, from which anybody can
prove anything at all.

Let us pretend again that the universe has NKS rudiments. Let us assume that the human thinking within the universe is also a product of rudimentary NKS (simple programs, algorithmic, etc.).

Well apparently, the rudiments allowed the universe to "step" to this new universe configuration in which the above assumption, which was false, was accepted.

The act of accepting a false assumption is apparently allowed by the algorithmic rudiments.

Posted by: Jon Awbrey

ALNP. Discussion Note 1

JA = Jon Awbrey
PD = Philip Dutton

JA: According to my understanding of it,
the so-called Liar Paradox is just the
most simple-minded of fallacies, involving
nothing more mysterious than the acceptance
of a false assumption, from which anybody can
prove anything at all.

PD: Let us pretend again that the universe has NKS rudiments.
Let us assume that the human thinking within the universe
is also a product of rudimentary NKS (simple programs,
algorithmic, etc.)

I think I've heard this one before --

| If we pretend that a tail is a leg,
| then how many legs does a dog have?

As good as I am at pretending, this would be a difficult pretense,
even for me. The way I pretend to understand it, the universe of
percourse that we drum up when we talk of "algorithmic rudiments",
and the whole repertoire of flim-flam paradigm-a-doodles that go
along with it, is just the space of recursive partial functions,
and last I counted there were only a countable number of these.
So the "universe at large" (UAL), unless it turns out to be
a "very large but still finite automaton" (VLBSFA), is just
sure to have all sorts of non-algorithmic happenings in it.

But even Aristotle already grasped the circumstance that
not all happenings in the world of phenomena fall within
the purview of science, but only the happenings that are,
as we say, "goings-on", in the sense of having a general
distribution throughout the experience of a community of
inquiry, in particular, persisting without limit in time.

Every method has a limit, indeed, the limit that makes it a method.
The more that one peers at the respective boundaries and interiors,
the more it begins to appear that the same limitation that gives a
method to science could very well be the same limitation to finite
means that stakes out the horizon of the computable.

To make a long story short, there is a sense of pretending that I can
entertain following directions with here, the sense in which a bit of
play-acting on some formal stage or other provides us with a model of
what goes on in the world outside the stage doors. But that requires
intelligent interpretation, not to mention cognizing the abbreviation,
the bias, the compression, and the distortion that is part and parcel
to the relationship between a reality and its finitary representation.

Jon Awbrey

PD: Well apparently, the rudiments allowed the universe
to "step" to this new universe configuration in which
the above assumption, which was false, was accepted.

PD: The act of accepting a false assumption
is apparently allowed by the algorithmic
rudiments.

Posted by: Gunnar Tomasson

Re. the following:

JA: According to my understanding of it,
the so-called Liar Paradox is just the
most simple-minded of fallacies, involving
nothing more mysterious than the acceptance
of a false assumption, from which anybody can
prove anything at all.

Question:

What is a "false assumption"?

Posted by: Jon Awbrey

ALNP. Discussion Note 2

Gunnar,

In this particular case, the false assumption was the
proposition "Statement_1 <=> Not Statement_1", that
I labeled as "Statement_0", and that contradicts
an axiom of logic.

Jon Awbrey

Posted by: Gunnar Tomasson

Jon:

From the vantage point of one interested in the epistemological aspects of modern physical science, the concept of a "false assumption" is problematic.

The nature of the problem is reflected in Einstein's statement in August 1954 that "it is quite possible that physics cannot be founded on the concept of field - that is to say, on continuous elements."

Here, the concept of "field" represents (a) an "assumption" about structural aspects of physical reality, and (b) a mathematical given for contemporary theoretical models thereof, where (b) is to (a) as "map" is to "territory".

But, as indicated by Einstein's remark, it is quite possible for an "assumption" to be "true" with respect to theoretical models of physical reality (b) and "false" with respect to such reality itself (a).

That is to say, an "assumption" - axiom - may be consistent with respect to (b) and inconsistent with respect to (a).

A case in point.

In contemporary physics, the concept of Black Hole grew out of the "assumption" that "physics can be founded on the concept of field - that is to say, on continuous elements".

And, while physicists will concede that, in principle, their theories can never be 'proved' but only 'falsified', in practice their modus operandi is such that it is logically impossible to 'falsify', say, the Black Hole theory because it is always and necessarily consistent or "true" with respect to its underlying "assumptions".

Einstein incurred the wrath of his peers by making the like point with respect to their Quantum Mechanical orthodoxy - and, as an intellectual outcast within the community of theoretical physicists for the last three decades of his life, he called their spade a spade in 1949 as follows:

"Science without epistemology is - insofar as it is thinkable at all - primitive and muddled."

Gunnar

Posted by: Jon Awbrey

ALNP. Discussion Note 3

Gunnar,

Just headed off to dreamland, where I usually
do all my best thinking, and so I will have to
sleep on this question for now.

But let me just mention the distinction between
descriptive sciences, like physics or psychology,
and normative sciences, like aesthetics, ethics,
and logic.

The so-called Liar Paradox is normally presented
as a difficulty within purely classical logic,
and so I analyzed it in that context.

This is a very different matter from the
approximate and even desirably defeasible
character of all contingent empirical laws.

(Tomorrow &)* ...

Jon Awbrey

Posted by: Jon Awbrey

ALNP. Note 2

| Algebraic Calculation
|
| For algebras, two rules
| are commonly accepted
| as implicit in the
| use of the sign =.
|
| Rules of Substitution and Replacement
|
| Rule 1. Substitution
|
| If e = f, and if h is an expression constructed
| by substituting f for any appearance of e in g,
| then g = h.
|
| Rule 2. Replacement
|
| If e = f, and if every token of a given independent
| variable expression v in e = f is replaced by an
| expression w, it not being necessary for v, w
| to be equivalent or for w to be independent
| or variable, and if as a result of this
| procedure e becomes j and f becomes k,
| then j = k.
|
| George Spencer Brown, 'Laws of Form',
| George Allen & Unwin, London, UK, 1969,
| combining texts at pp. 26-27 and p. 140.

Posted by: Jon Awbrey

ALNP. Note 3

| As far as the laws of mathematics refer to reality, they are not
| certain; and as far as they are certain, they do not refer to
| reality. It seems to me that complete clearness as to this
| state of things first became common property through that
| new departure in mathematics which is known by the name
| of mathematical logic or "Axiomatics". The progress
| achieved by axiomatics consists in its having neatly
| separated the logical-formal from its objective or
| intuitive content.
|
| Albert Einstein, "Geometry and Experience" (1921),
| in 'Sidelights on Relativity', Dover, 1983, p. 28-29.
|
| http://www.bun.kyoto-u.ac.jp/~suchii/EonGeometry.html

Posted by: Jon Awbrey

ALNP. Note 4

| Matter is potentiality (dynamis), while form is
| realization or actuality (entelecheia), and the
| word actuality is used in two senses, illustrated
| by the possession of knowledge (episteme) and the
| exercise of it (theorein).
|
| So the soul (psyche) must be substance (ousia)
| in the sense of being the form (eidos) of a
| natural body (soma), which potentially (dynamei)
| has life (zoe). And substance in this sense is
| actuality (entelecheia).
|
| Aristotle, "Peri Psyche", 2.1.

| The passage from power to entelechy takes place
| by means of change (kinesis). This is the
| imperfect energy, the perfected energy
| is the entelechy.
|
| C.S. Peirce, 'Chronological Edition', CE 5, p. 404.

| I shall, therefore, venture to call [Sum (mv^2)/2]
| the kinetic act or kinetic energy, and the negative
| of the potential, the kinetic power or kinetic
| potency. For the sum of the two I can think of
| no better term than 'motivity' or 'kinesis'.
|
| C.S. Peirce, 'Chronological Edition', CE 5, p. 275n.

| Tho' obscur'd, this is the form of the Angelic land.
|
| William Blake, "America",
| Inductory Envoi to the American Edition of:
| George Spencer Brown, 'Laws of Form', 1972.

| In arriving at proofs, I have often been struck
| by the apparent alignment of mathematics with
| psycho-analytic theory. In each discipline
| we attempt to find out, by a mixture of
| contemplation, symbolic representation,
| communion, and communication, what
| it is we already know.
|
| George Spencer Brown, 'Laws of Form', p. xix.

| One of the motives prompting the furtherance of
| the present work was the hope of bringing together
| the investigations of the inner structure of our
| knowledge of the universe, as expressed in the
| mathematical sciences, and the investigations of
| its outer structure, as expressed in the physical
| sciences. Here the work of Einstein, Schrodinger,
| and others seems to have led to the realization of
| an ultimate boundary of physical knowledge in the
| form of the media through which we perceive it.
|
| George Spencer Brown, 'Laws of Form', p. xxi.

| What is encompassed, in mathematics, is a
| transcedence from a given state of vision to a
| new, and hitherto unapparent, vision beyond it.
| When the present existence has ceased to make
| sense, it can still come to sense again through
| the realization of its form.
|
| George Spencer Brown, 'Laws of Form', p. xxiii.
|
| [ Yes, it's "transcedence", not "transcendence"!!! ]
| [ Sic transit gloria mundi, what a diff "n" makes! ]

| One of the most beautiful facts emerging from
| mathematical studies is this very potent relation-
| ship between the mathematical process and ordinary
| language. There seems to be no mathematical idea
| of any importance or profundity that is not
| mirrored, with an almost uncanny accuracy, in the
| common use of words, and this appears especially
| true when we consider words in their original,
| and sometimes long forgotten, senses.
|
| George Spencer Brown, 'Laws of Form', pp. 90-91.

| Thus we do not imagine the wave train emitted by
| an excited finite echelon to be exactly like the
| wave train emitted from an excited physical
| particle. For one thing the wave form from an
| echelon is square, and for another it is emitted
| without energy. (We should need, I guess, to make
| at least one more departure from the form before
| arriving at a conception of energy on these lines.)
|
| George Spencer Brown, 'Laws of Form', p. 100.

| Ladies and gentlemen, I have set this aspect
| of exact science before you because in it the
| affinity with the fine arts becomes most plainly
| visible, and because here one may counter the
| misapprehension that natural science and
| technology are concerned solely with precise
| observation and rational, discursive thought.
| To be sure, this rational thinking and careful
| measurement belong to the scientist's work, just
| as the hammer and chisel belong to the work of the
| sculptor. But in both cases they are merely the
| tools and not the content of the work.
|
| Werner Heisenberg,
|"The Meaning of Beauty in the Exact Sciences",
|'Across the Frontiers', p. 182.